Is there an example of a Markov semigroup $(P_t)_{t \geq 0}$ on a measurable space $(E, \mathcal{E})$ that does not appear as a transition semigroup of a Markov process $(X_t)_{t \geq 0}$ on $(E, \mathcal{E})$?
It is known that if $E$ is a standard Borel space (i.e. a measurable subset of a Polish space) then for any Markov semigroup on $E$ (together with an initial distribution) one can define a Markov process (which then has the given Markov semigroup as its transition semigroup). This is true due to Kolmogorov's theorem. So for a counterexample, $E$ should be necessarily not a standard Borel space.